Bifurcation for a Class of Indefinite Elliptic Systems by Comparison Theory for the Spectral Flow via an Index Theorem

TL;DR

This paper develops a new comparison principle based on spectral flow and index theory to analyze bifurcation in indefinite elliptic PDE systems, with applications to bifurcation on shrinking domains.

Contribution

It introduces a refined comparison principle using spectral flow and index theorems for Fredholm operators, advancing bifurcation analysis methods for indefinite elliptic systems.

Findings

Established a new bifurcation criterion for indefinite elliptic systems.

Applied the theory to bifurcation problems on shrinking domains.

Extended previous comparison principles with spectral flow techniques.

Abstract

We consider families of strongly indefinite systems of elliptic PDE and investigate bifurcation from a trivial branch of solutions by using the spectral flow. The novelty in our approach is a refined version of a comparison principle that was originally proved by Pejsachowicz in a joint work with the third author, and which is based on an index theorem for a certain class of Fredholm operators that is of independent interest. Finally, we use our findings for a bifurcation problem on shrinking domains that originates from works of Morse and Smale.